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Theorem 1stf2 16814
 Description: Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfval.t 𝑇 = (𝐶 ×c 𝐷)
1stfval.b 𝐵 = (Base‘𝑇)
1stfval.h 𝐻 = (Hom ‘𝑇)
1stfval.c (𝜑𝐶 ∈ Cat)
1stfval.d (𝜑𝐷 ∈ Cat)
1stfval.p 𝑃 = (𝐶 1stF 𝐷)
1stf1.p (𝜑𝑅𝐵)
1stf2.p (𝜑𝑆𝐵)
Assertion
Ref Expression
1stf2 (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))

Proof of Theorem 1stf2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 1stfval.b . . . 4 𝐵 = (Base‘𝑇)
3 1stfval.h . . . 4 𝐻 = (Hom ‘𝑇)
4 1stfval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 1stfval.d . . . 4 (𝜑𝐷 ∈ Cat)
6 1stfval.p . . . 4 𝑃 = (𝐶 1stF 𝐷)
71, 2, 3, 4, 5, 61stfval 16812 . . 3 (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
8 fo1st 7173 . . . . . 6 1st :V–onto→V
9 fofun 6103 . . . . . 6 (1st :V–onto→V → Fun 1st )
108, 9ax-mp 5 . . . . 5 Fun 1st
11 fvex 6188 . . . . . 6 (Base‘𝑇) ∈ V
122, 11eqeltri 2695 . . . . 5 𝐵 ∈ V
13 resfunexg 6464 . . . . 5 ((Fun 1st𝐵 ∈ V) → (1st𝐵) ∈ V)
1410, 12, 13mp2an 707 . . . 4 (1st𝐵) ∈ V
1512, 12mpt2ex 7232 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V
1614, 15op2ndd 7164 . . 3 (𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
177, 16syl 17 . 2 (𝜑 → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
18 simprl 793 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑥 = 𝑅)
19 simprr 795 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑦 = 𝑆)
2018, 19oveq12d 6653 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2120reseq2d 5385 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (1st ↾ (𝑥𝐻𝑦)) = (1st ↾ (𝑅𝐻𝑆)))
22 1stf1.p . 2 (𝜑𝑅𝐵)
23 1stf2.p . 2 (𝜑𝑆𝐵)
24 ovex 6663 . . . 4 (𝑅𝐻𝑆) ∈ V
25 resfunexg 6464 . . . 4 ((Fun 1st ∧ (𝑅𝐻𝑆) ∈ V) → (1st ↾ (𝑅𝐻𝑆)) ∈ V)
2610, 24, 25mp2an 707 . . 3 (1st ↾ (𝑅𝐻𝑆)) ∈ V
2726a1i 11 . 2 (𝜑 → (1st ↾ (𝑅𝐻𝑆)) ∈ V)
2817, 21, 22, 23, 27ovmpt2d 6773 1 (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  Vcvv 3195  ⟨cop 4174   ↾ cres 5106  Fun wfun 5870  –onto→wfo 5874  ‘cfv 5876  (class class class)co 6635   ↦ cmpt2 6637  1st c1st 7151  2nd c2nd 7152  Basecbs 15838  Hom chom 15933  Catccat 16306   ×c cxpc 16789   1stF c1stf 16790 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-hom 15947  df-cco 15948  df-xpc 16793  df-1stf 16794 This theorem is referenced by:  1stfcl  16818  prf1st  16825  1st2ndprf  16827  uncf2  16858  diag12  16865  diag2  16866  yonedalem22  16899
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